A large industrial firm allows a discount on any in voice that is paid within 30

Question

A large industrial firm allows a discount on any in voice that is paid within 30 days. Of all invoices, 10% receive the discount. In a company audit, 12 invoices are sampled at random. a. What is the probability that exactly four of them receive the discount? b. What is the probability that fewer than three of hem receive the discount? c. What is the probability that none of them receive the discount? d. Find the mean number that receive the discount. e. Find the standard deviation of the number that receive the discount 7. A fair coin is tossed eight times.

Explanation / Answer

(6)

The probability of receiving discount is given as 0.1

Modeling this as a Binomial distribution with the following parameters:

n = 12, p = 0.10

Let X denote the number of invoices that receive the discount.

(a)

P(exactly four receive the discount) = P(X = 4)

Using the CDF for a binomial distribution:

P(X=x) = nCx*(p^x)*((1-p)^(n-x))

P(X=4) = 12C4*(0.1^4)*((1-0.1)^(12-4)) = 0.021

(b)

P(fewer than three receive the discount) = P(X < 3) = P(X=0) + P(X=1) + P(X=2)

Now,

P(X=0) = 12C0*(0.1^0)*((1-0.1)^(12-0)) = 0.282

P(X=1) = 12C1*(0.1^1)*((1-0.1)^(12-1)) = 0.376

P(X=2) = 12C2*(0.1^2)*((1-0.1)^(12-2)) = 0.23

So,

P(X < 3) = 0.282+0.376+0.23 = 0.888

(c)

P(none of them receive the discount) = P(X=0)

As calculated as above,

P(X=0) = 0.282

(d)

For a Binomial distribution:

Mean = n*p = 12*0.1 = 1.2

(e)

For a Binomial distribution:

Standard Deviation = (n*p*(1-p))^0.5 = (12*0.1*(1-0.1))^0.5 = 1.039

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